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The Trapezoidal Rule

The trapezoidal rule is a simple average of the forward-Euler and backward-Euler schemes. It can be shown that the local truncation error scales with $ h^3$.

$\displaystyle y(t+h) = y(t) + \frac{h \cdot (y^{(1)}(t) + y^{(1)}(t+h))}{2}$ (1.6)

For ordinary differential equations, the trapezoidal rule is an application of the $ \theta$ method, which itself is a special case of a second-order Runge-Kutta method. For more details see [6].

Figure 1.3: Graphical illustration of the trapezoidal method. Starting at point 1, we get point 2 by taking the derivatives at point 1 and point 2, and extrapolating their average in point 1.
\includegraphics[scale=0.6]{figures/trapez.eps}


Table 1.1: The numerical schemes of interest. Not all of them are mentioned in the text, see [8] and [6] for more information.
  Explicit Implicit

First Order

Forward-Euler, exponential Euler Backward-Euler
Second Order 2nd-Order Runge-Kutta Trapezoidal rule
Higher Order Adams-Bashforth, Runge-Kutta-Fehlberg Adams-Moulton, Gear



next up previous contents
Next: Neuroscience Equations Up: Numerical Schemes Previous: Exponential Euler   Contents
2002-11-15